Communications Project

Document Type:Dissertation
Name:Andrew S. Repp
Title:Discrete Riemann Maps and the Parabolicity of Tilings
Degree:Doctor of Philosophy
Committee Chair: William J. Floyd
Committee Members:Peter E. Haskell
Peter A. Linnell
Robert A. McCoy
James E. Thomson
Keywords:Tilings, Parabolic, Modulus, Riemann Map
Date of defense:May 4, 1998
Availability:Release the entire work for Virginia Tech access only.
After one year release worldwide only with written permission of the student and the advisory committee chair.


The classical Riemann Mapping Theorem has many discrete analogues. One of these, the Finite Riemann Mapping Theorem of Cannon, Floyd, Parry,and others, describes finite tilings of quadrilaterals and annuli. It relates to several combinatorial moduli, similar in nature to the classical modulus. The first chapter surveys some of these discrete analogues. The next chapter considers appropriate extensions to infinite tilings of half-open quadrilaterals and annuli. In this chapter we prove some results about combinatorial moduli for such tilings. The final chapter considers triangulations of open topological disks. It has been shown that one can classify such triangulations as either parabolic or hyperbolic, depending on whether an associated combinatorial modulus is infinite or finite. We obtain a criterion for parabolicity in terms of the degrees of vertices that lie within a specified distance of a given base vertex.

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